### 18.11 Other random number generators

The generators in this section are provided for compatibility with existing libraries. If you are converting an existing program to use GSL then you can select these generators to check your new implementation against the original one, using the same random number generator. After verifying that your new program reproduces the original results you can then switch to a higher-quality generator.

Note that most of the generators in this section are based on single linear congruence relations, which are the least sophisticated type of generator. In particular, linear congruences have poor properties when used with a non-prime modulus, as several of these routines do (e.g. with a power of two modulus, 2^31 or 2^32). This leads to periodicity in the least significant bits of each number, with only the higher bits having any randomness. Thus if you want to produce a random bitstream it is best to avoid using the least significant bits.

Generator: gsl_rng_ranf

This is the CRAY random number generator `RANF`. Its sequence is

```x_{n+1} = (a x_n) mod m
```

defined on 48-bit unsigned integers with a = 44485709377909 and m = 2^48. The seed specifies the lower 32 bits of the initial value, x_1, with the lowest bit set to prevent the seed taking an even value. The upper 16 bits of x_1 are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc. produce the same sequences.

The generator compatible with the CRAY MATHLIB routine RANF. It produces double precision floating point numbers which should be identical to those from the original RANF.

There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation.

Note that you can only seed the generator with integers up to 2^32, while the original CRAY implementation uses non-portable wide integers which can cover all 2^48 states of the generator.

The function `gsl_rng_get` returns the upper 32 bits from each term of the sequence. The function `gsl_rng_uniform` uses the full 48 bits to return the double precision number x_n/m.

The period of this generator is 2^46.

Generator: gsl_rng_ranmar

This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman and Tsang. It is a 24-bit generator, originally designed for single-precision IEEE floating point numbers. It was included in the CERNLIB high-energy physics library.

Generator: gsl_rng_r250

This is the shift-register generator of Kirkpatrick and Stoll. The sequence is based on the recurrence

```x_n = x_{n-103} ^^ x_{n-250}
```

where ^^ denotes “exclusive-or”, defined on 32-bit words. The period of this generator is about 2^250 and it uses 250 words of state per generator.

• S. Kirkpatrick and E. Stoll, “A very fast shift-register sequence random number generator”, Journal of Computational Physics, 40, 517–526 (1981)
Generator: gsl_rng_tt800

This is an earlier version of the twisted generalized feedback shift-register generator, and has been superseded by the development of MT19937. However, it is still an acceptable generator in its own right. It has a period of 2^800 and uses 33 words of storage per generator.

• Makoto Matsumoto and Yoshiharu Kurita, “Twisted GFSR Generators II”, ACM Transactions on Modelling and Computer Simulation, Vol. 4, No. 3, 1994, pages 254–266.
Generator: gsl_rng_vax

This is the VAX generator `MTH\$RANDOM`. Its sequence is,

```x_{n+1} = (a x_n + c) mod m
```

with a = 69069, c = 1 and m = 2^32. The seed specifies the initial value, x_1. The period of this generator is 2^32 and it uses 1 word of storage per generator.

Generator: gsl_rng_transputer

This is the random number generator from the INMOS Transputer Development system. Its sequence is,

```x_{n+1} = (a x_n) mod m
```

with a = 1664525 and m = 2^32. The seed specifies the initial value, x_1.

Generator: gsl_rng_randu

This is the IBM `RANDU` generator. Its sequence is

```x_{n+1} = (a x_n) mod m
```

with a = 65539 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator was only 2^29. It has become a textbook example of a poor generator.

Generator: gsl_rng_minstd

This is Park and Miller’s “minimal standard” MINSTD generator, a simple linear congruence which takes care to avoid the major pitfalls of such algorithms. Its sequence is,

```x_{n+1} = (a x_n) mod m
```

with a = 16807 and m = 2^31 - 1 = 2147483647. The seed specifies the initial value, x_1. The period of this generator is about 2^31.

This generator was used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function) in the past. It is also sometimes known by the acronym “GGL” (I’m not sure what that stands for).

• Park and Miller, “Random Number Generators: Good ones are hard to find”, Communications of the ACM, October 1988, Volume 31, No 10, pages 1192–1201.
Generator: gsl_rng_uni
Generator: gsl_rng_uni32

This is a reimplementation of the 16-bit SLATEC random number generator RUNIF. A generalization of the generator to 32 bits is provided by `gsl_rng_uni32`. The original source code is available from NETLIB.

Generator: gsl_rng_slatec

This is the SLATEC random number generator RAND. It is ancient. The original source code is available from NETLIB.

Generator: gsl_rng_zuf

This is the ZUFALL lagged Fibonacci series generator of Peterson. Its sequence is,

```t = u_{n-273} + u_{n-607}
u_n  = t - floor(t)
```

The original source code is available from NETLIB. For more information see,

• W. Petersen, “Lagged Fibonacci Random Number Generators for the NEC SX-3”, International Journal of High Speed Computing (1994).
Generator: gsl_rng_knuthran2

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

```x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
```

with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.

Generator: gsl_rng_knuthran2002
Generator: gsl_rng_knuthran

This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth provides its C code. The updated routine `gsl_rng_knuthran2002` is from the revised 9th printing and corrects some weaknesses in the earlier version, which is implemented as `gsl_rng_knuthran`.

Generator: gsl_rng_borosh13
Generator: gsl_rng_fishman18
Generator: gsl_rng_fishman20
Generator: gsl_rng_lecuyer21
Generator: gsl_rng_waterman14

These multiplicative generators are taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Their sequence is,

```x_{n+1} = (a x_n) mod m
```

where the seed specifies the initial value, x_1. The parameters a and m are as follows, Borosh-Niederreiter: a = 1812433253, m = 2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271, m = 2^31 - 1, L’Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a = 1566083941, m = 2^32.

Generator: gsl_rng_fishman2x

This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,

```z_{n+1} = (x_n - y_n) mod m
```

with m = 2^31 - 1. x_n and y_n are given by the `fishman20` and `lecuyer21` algorithms. The seed specifies the initial value, x_1.

Generator: gsl_rng_coveyou

This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence is,

```x_{n+1} = (x_n (x_n + 1)) mod m
```

with m = 2^32. The seed specifies the initial value, x_1.